arXiv subjects

Wenliang Zhang

Publications and source records attributed to Wenliang Zhang.

94 records · Page 6Linked to original sources

arXiv2008

On the discreteness and rationality of F-jumping coefficients

This paper studies the jumping coefficients of principal ideals of regular local rings. Recently M. Blickle, M. Mustata and K. Smith showed that, when $R$ is of essentially finite type over a field and $F$-finite, bounded intervals contain finitely many jumping coefficients and that those are rational. In a later paper they extended these results to principal ideals of $F$-finite complete regular local rings. The aim of this paper is to extend these results on the discreteness and rationality of jumping coefficients to principal ideals of arbitrary (i.e. not necessarily $F$-finite) excellent regular local rings containing fields of positive characteristic. Our proof uses a very different method: we do not use $D$-modules and instead we analyze the modules of nilpotents elements in the injective hull or $R$ under some non-standard Frobenius actions. This new method undoubtedly holds a potential for more applications.

math.AC↗

arXiv2007

On the Frobenius functor and colon ideals

In this paper we study the commutativity of the Frobenius functor and the colon operation of two ideals for Noetherian rings of positive characteristic $p$. New characterizations of regular rings and local UFDs are given.

math.AC↗

arXiv2007

On Lyubeznik numbers of projective schemes

Let $X$ be an arbitrary projective scheme over a field $k$. Let $A$ be the local ring at the vertex of the affine cone for some embedding $ι: X\hookrightarrow \mathbb{P}^n_k$. G. Lyubeznik asked (in \cite{l2}) whether the integers $λ_{i,j}(A)$ (defined in \cite{l1}), called the Lyubeznik numbers of $A$, depend only on $X$, but not on the embedding. In this paper, we make a big step toward a positive answer to this question by proving that in positive characteristic, for a fixed $X$, the Lyubezink numbers $λ_{i,j}(A)$ of the local ring $A$, can only achieve finitely many possible values under all choices of embeddings.

math.AC↗

arXiv2006

On the highest Lyubeznik number of a local ring

Let $A$ be a $d$-dimensional local ring containing a field. We will prove that the highest Lyubeznik number $λ_{d,d}(A)$ (defined in \cite{l1}) is equal to the number of connected components of the Hochster-Huneke graph (defined in \cite{hh}) associated to $B$, where $B=\hat{\hat{A}^{sh}}$ is the completion of the strict Henselization of the completion of $A$. This was proven by Lyubeznik in characteristic $p>0$. Our statement and proof are characteristic-free.

math.AC↗